Question 195572
{{{(x+3)/(x-5)=(56-3x)/(x^2-13x+40)}}} Start with the given equation.



{{{(x+3)/(x-5)=(56-3x)/((x-5)(x-8))}}} Factor



{{{cross((x-5))(x-8)((x+3)/cross((x-5)))=cross((x-5)(x-8))((56-3x)/cross((x-5)(x-8)))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{(x-5)(x-8)}}}. Doing this will eliminate all of the fractions.




{{{(x-8)(x+3)=56-3x}}} Cancel out and simplify.



{{{x^2-5x-24=56-3x}}} FOIL



{{{x^2-5x-24-56+3x=0}}} Get all terms to the left side.



{{{x^2-2x-80=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-2}}}, and {{{c=-80}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-2) +- sqrt( (-2)^2-4(1)(-80) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-2}}}, and {{{c=-80}}}



{{{x = (2 +- sqrt( (-2)^2-4(1)(-80) ))/(2(1))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{x = (2 +- sqrt( 4-4(1)(-80) ))/(2(1))}}} Square {{{-2}}} to get {{{4}}}. 



{{{x = (2 +- sqrt( 4--320 ))/(2(1))}}} Multiply {{{4(1)(-80)}}} to get {{{-320}}}



{{{x = (2 +- sqrt( 4+320 ))/(2(1))}}} Rewrite {{{sqrt(4--320)}}} as {{{sqrt(4+320)}}}



{{{x = (2 +- sqrt( 324 ))/(2(1))}}} Add {{{4}}} to {{{320}}} to get {{{324}}}



{{{x = (2 +- sqrt( 324 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 18)/(2)}}} Take the square root of {{{324}}} to get {{{18}}}. 



{{{x = (2 + 18)/(2)}}} or {{{x = (2 - 18)/(2)}}} Break up the expression. 



{{{x = (20)/(2)}}} or {{{x =  (-16)/(2)}}} Combine like terms. 



{{{x = 10}}} or {{{x = -8}}} Simplify. 



So the answers are {{{x = 10}}} or {{{x = -8}}}